Probability Abstracts 51

1280. IMPROVED BONFERRONI INEQUALITIES VIA UNION-CLOSED SET SYSTEMS

K. Dohmen

By applying a recent result of Naiman and Wynn on abstract tubes, we
establish a new improvement of the classical Bonferroni inequalities for
any finite and indexed collection of events having an additional
structure, which is assumed to be given in terms of a union-closed set
of non-empty subsets of the index set. The result generalizes several
other results from the literature.

dohmen@informatik.hu-berlin.de

1281. THE RANDOM GEOMETRY OF EQUILIBRIUM PHASES

Hans-Otto Georgii, Olle Haggstrom and Christian Maes

This is a large (118 pages) survey about applications of
percolation theory in equilibrium statistical mechanics. The
chapters are as follows:
1. Introduction 
2. Equilibrium phases 
3. Some models
4. Coupling and stochastic domination 
5. Percolation
6. Random-cluster representations
7. Uniqueness and exponential mixing from non-percolation 
8. Phase transition and percolation
9. Random interactions
10. Continuum models

georgii@rz.mathematik.uni-muenchen.de
olleh@math.chalmers.se
Christian.Maes@fys.kuleuven.ac.be

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1282. THE GEOMETRY OF MARKOV DIFFUSION GENERATORS

M. Ledoux

These notes form a summary of a mini-course given at the Eidgen\"ossische
Technische Hochschule in Z\"urich in November 1998. They aim to present
some of the basic ideas in the geometric investigation of Markov diffusion
generators, as developed during the last decade by D. Bakry and his
collaborators. In particular, abstract notions of curvature and dimension
that extend the corresponding ones in Riemannian geometry are introduced
and studied. Using functional tools, new analytic proofs of some classical
comparison theorems in Riemannian geometry are presented together with
their counterparts in infinite dimension. Particular emphasis is put on
sharp constants, optimal inequalities and model spaces and generators.
These notes
are far from complete (in particular most proofs are only outlined) and only
aim to give a flavour of the subject.

ledoux@cict.fr (ps file available)

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1283. LARGE DEVIATIONS AND VARIATIONAL PRINCIPLE FOR HARMONIC CRYSTALS

P. Caputo and J.-D. Deuschel

We consider massless Gaussian fields with covariance
related to the Green function of a long range random walk on $\bbZ^d$.
These are viewed as Gibbs measures for a 
linear-quadratic interaction. 
We establish thermodynamic identities and prove a
version of Gibbs' variational principle, showing that
translation invariant Gibbs measures are characterized as
minimizers of the relative entropy density.  
We then study the large deviations of the empirical field 
of a Gibbs measure. We show that a weak
large deviation principle holds at the volume order,
with rate given by the relative entropy density. 

caputo@math.TU-Berlin.DE

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1284. UPPER BOUNDS ON THE HEIGHT DIFFERENCE OF THE GAUSSIAN RANDOM FIELD AND THE RANGE OF RANDOM GRAPH HOMOMORPHISM INTO Z

Itai Benjamini and Gideon Schechtman

It is shown that the maximal height difference of the Gaussian random
field on a connected graph with
n vertices is dominated by

\max_{1 \leq j <k \leq 2n} |\sum_{i=j}^k X_i|.

Where the X_i's, 1 \leq i \leq 2n,
are independent standard normal random variables.
A somewhat weaker analogue of this, is established for the
range of random graph homomorphisms into Z, 
completing a conjecture from  Benjamini, H\"aggstr\"om and
Mossel (1998).

itai@wisdom.weizmann.ac.il

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1285. REFLECTION AND COALESCENCE BETWEEN INDEPENDENT ONE-DIMENSIONAL BROWNIAN PATHS

Florin Soucaliuc, Balint Toth, Wendelin Werner

Take two independent one-dimensional
Brownian motions $(B_t , t \in [0,1])$
and $(\beta_t , t \in [0,1])$ with $B_0= 0$ and $\beta_1= 0$
($\beta$ can be seen as running  backwards in time).
Define $(\gamma_t , t \in [0,1])$
as being the function that is obtained by reflecting $B$ on $\beta$.
Then $\gamma$ is still a Brownian motion.
Similar and more general results (with families
of coalescing Brownian motions) are also derived.
They enable to give a precise definition (in terms of reflection)
of the joint realisation of finite families 
of coalescing/reflecting brownian motions.

florin.soucaliuc@math.u-psud.fr
balint@math-inst.hu
wendelin.werner@math.u-psud.fr

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1286. TIGHTNESS OF LOCALIZATION AND RETURN TIME IN RANDOM ENVIRONMENT

Yueyun Hu

Consider a class of diffusions with  random potentials 
which asymptotically behave as a Brownian motion, we study 
the tightness of  localization around the bottom of some 
Brownian valley, and determine the limit distribution of 
their return time to the origin after a typical time. 
Via  the Skorokhod embedding  in  random environment, 
we also solve the return time problem for Sinai's walk. 

hu@ccr.jussieu.fr

1287. A LEVEQUE-TYPE LOWER BOUND FOR DISCREPANCY

Francis Edward Su

A sharp lower bound for discrepancy on $\R/\Z$ is derived
which resembles the upper bound due to LeVeque.  An 
analogous bound is proved for discrepancy on $\R^k/\Z^k$.  
These are discussed in the more general context of the 
discrepancy of probability measures.  As applications, 
the bounds are applied to Kronecker sequences and to a 
random walk on the torus.

su@math.hmc.edu

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1288. RANDOM WALKS WITH BADLY APPROXIMABLE NUMBERS

Doug Hensley and Francis Edward Su

We analyze the rate of convergence of a random walk on
the circle generated by $d$ rotations, and establish sharp
rates that show that badly approximable $d$-tuples in 
$\R^d$ give rise to walks with the fastest convergence.  
For the case $d=2$ we exhibit what we believe is the 
optimal pair of generators, and show that it is near 
optimal.

su@math.hmc.edu

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1289. ARBITRAGE FREE VALUATION OF A FEDERAL TIMBER LEASE

Ellen Burnes, Enrique Thomann and Ed Waymire

The objective of this paper is to describe a quantitative 
framework for offering timber harvest bids on federal lands 
which includes special considerations of the volatility of 
timber indices and the harvesting costs involved in 
harvesting timber.  The advantage of such an approach is 
that it provides a precise framework in which various 
underlying considerations, such as volatility and cost, may 
be systematically defined, measured and evaluated.  
The valuations derived in this paper provide a market 
standard against which additional value that encompasses 
social or environmental welfare may be evaluated. 
We discuss a specific USDA Forest Service timber sale
as a case study to illustrate this approach.

burnese@ucs.orst.edu, thomann@math.orst.edu, waymire@math.orst.edu

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1290. PERCOLATION ON NONAMENABLE PRODUCTS AT THE UNIQUENESS THRESHOLD

Yuval Peres

Let $X$ and $Y$ be infinite quasi-transitive graphs,
such that the automorphism group of $X$ is not
amenable. For i.i.d. percolation on the Cartesian
product $X \times Y$, we show that the set of
retention parameters $p$ where a.s. there is a unique
infinite cluster, does not contain its infimum $p_u$.
This extends a result of Schonmann, who considered
the Cartesian product of a regular tree and the integers.

peres@math.huji.ac.il

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1291. UNIVERSALITY FOR CONFORMALLY INTERSECTION EXPONENTS

Greg Lawler and Wendelin Werner

We construct a class of conformally 
invariant  measures on sets (or  paths) and we
study the critical exponents called intersection
exponents associated to these measures. 
we show that these exponents exist and that they  correspond
to intersection exponents between planar  Brownian
motions. More precisely, using the definitions 
and results of a former paper of ours,
we show that any set defined
under such a conformal invariant measure 
behaves exactly as a pack
(containing maybe a non-integer number) of Brownian motions
as what all intersection exponents are concerned.
We show how conjectures about exponents for
two-dimensional self-avoiding
walks and critical percolation clusters can be reinterpreted
in terms of conjectures on Brownian exponents.

jose@math.duke.edu  wendelin.werner@math.u-psud.fr

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1292. THICK POINTS FOR PLANAR BROWNIAN MOTION AND THE ERDOS-TAYLOR CONJECTURE ON RANDOM WALK

Amir Dembo, Yuval Peres, Jay Rosen and Ofer Zeitouni

Denote by $T(x,r)$ the occupation measure of a disc of 
radius $r$ around $x$ by planar Brownian motion run till 
time 1, and let $T(r)$ be the maximum of $T(x,r)$ over $x$ 
in the plane. We prove that $T(r)$ is a.s. asymptotic to 
$2 r^2 |\log r|^2$ as  $r$ tends to $0$, thus solving a 
problem posed by Perkins and Taylor (1987). Furthermore, 
for any $a<2$,  the Hausdorff dimension of the set of points 
$x$ for which $T(x,r)$ is asymptotic to $a r^2 |\log r|^2$, 
is almost surely $2-a$. As a consequence, we prove a 
conjecture about planar simple random walk due to Erdos and 
Taylor (1960): The number of visits to the most frequently 
visited lattice site in the first $n$ steps of the walk, is 
asymptotic to $(\log n)^2/\pi$. We also show that for 
positive $a<1/pi$,the number of points visited more than 
$a(\log n)^2$  times in the first $n$ steps, is 
approximately $n^{1-a\pi}$.

amir@stat.stanford.edu peres@math.huji.ac.il jrosen3@idt.net  zeitouni@ee.technion.ac.il

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1293. SELF-SIMILAR SETS OF ZERO HAUSDORFF MEASURE AND POSITIVE PACKING MEASURE

Yuval Peres, Karoly Simon and Boris Solomyak

We prove that there exist self-similar sets of zero 
Hausdorff measure, but positive and finite packing 
measure, in their dimension. For instance, if $U$ 
is picked uniformly at random in $[3,6]$, then the 
set of sums $\sum_n a_n 4^{-n}$ with $a_n$ in 
${0,1,U}$, has this property almost surely.

peres@math.huji.ac.il
simon@gold.uni-miskolc.hu
solomyak@math.washington.edu

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1294. PERCOLATION IN A DEPENDENT RANDOM ENVIRONMENT

Johan Jonasson, Elchanan Mossel and Yuval Peres

Draw planes in $\R^3$ that are orthogonal to the 
$z$ axis, and intersect that axis at the points 
of a Poisson process with intensity $\lambda$; 
similarly, draw planes orthogonal to the $x$ and 
$y$ axes using independent Poisson processes (with 
the same intensity) on these axes. This yields
a randomly stretched rectangular lattice. Consider 
bond percolation on this lattice where each edge of 
length $\ell$ is (independently) open with probability 
$e^{-\ell}$. We show that this model exhibits a phase 
transition: For large enough $\lambda$, there is an 
infinite open cluster a.s., and for small $\lambda$, all 
open clusters are finite a.s. (The question whether the 
analogous process in two dimensions exhibits a phase 
transition is open.) We prove this result using the 
method of `Paths With Exponential Intersection Tails'; 
This method yields a proof of phase transition for a 
large class of percolation processes in random environment 
in dimension $d \geq 3$.

jonasson@math.chalmers.se
mossel@math.huji.ac.il 
peres@math.huji.ac.il

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1295. GENERALIZATION OF IT\^O'S FORMULA FOR SMOOTH NONDEGENERATE MARTINGALES

Silvia Moret and David Nualart

In this paper we prove the existence of the quadratic 
covariations $[\frac{\partial F}{\partial x_k}(X),X^k]$, 
$1\le k\le d$, where $F$ belongs locally to the Sobolev
space ${\cal W}^{1,p}({\Bbb R}^d)$ for some $p>d$,
and $X$ is a $d$-dimensional smooth nondegenerate martingale
adapted to a $d$-dimensional Brownian motion. 
This result is based on some moment estimates 
for Riemann sums which are established by means of the 
techniques of the Malliavin calculus. As an application 
we establish an extension of It\^os formula for $F(X_t)$, 
where the complementary term is one half the sum of the 
above quadratic covariations.

moret@mat.ub.es nualart@mat.ub.es

1296. STOCHASTIC CALCULUS WITH RESPECT TO FRACTIONAL BROWNIAN MOTION WITH HURST PARAMETER LESSER THAN 1/2

Elisa Al\`os, Olivier Mazet and David Nualart

In this paper we introduce a stochastic integral with 
respect to the process $B_t=\int_0^t(t-s)^{-\alpha }dW_s$ 
where $0<\alpha <1/2$, and $W_t$ is a Brownian motion. 
Sufficient integrability conditions are deduced using the
techniques of the Malliavin calculus and the notion of 
fractional derivative. We study continuity properties of 
the indefinite integral and we derive a maximal inequality.

ealos@porthos.bio.ub.es mazet@mat.ub.es nualart@mat.ub.es

1297. A COST MODIFIED BLACK-SCHOLES-MERTON EQUATION

Enrique Thomann, Edward C. Waymire

The purpose of this paper is to introduce the mathematical
foundations of a new pricing formula for certain options 
found in the valuation of federal timber leases by Burnes, 
Thomann, Waymire (1999). The essential elements of such 
leases are a payoff rule based on the performance of some 
underlying timber values and the availability of this asset
in two forms, the raw or undeveloped state of standing timber
and the developed state of harvested timber. Implicit to 
this, therefore, is a conversion cost which must be reconciled
in the theory. We show that this framework leads naturally 
to the consideration of a mixture of Asian type European 
options with a down-and-out knock barrier.

thomann@math.orst.edu   waymire@math.orst.edu

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1298. SNAKES AND SPIDERS: BROWNIAN MOTION ON R-TREES

Steven N. Evans

We consider diffusion processes on a class of R-trees.
The processes are defined in a manner similar to that of 
Le Gall's Brownian snake.  Each point in the tree has a 
real--valued ``height'' or ``generation'',
and the height of the diffusion process evolves as a 
Brownian motion. When the height process decreases 
the diffusion retreats back along a lineage, whereas when 
the height process increases the diffusion chooses among 
branching lineages according to relative weights given by
a possibly infinite measure on the family of lineages.  
The class of R-trees we consider can have branch points
with countably infinite branching and lineages along which 
the branch points have points of accumulation.

We give a rigorous construction of the diffusion process, 
identify its Dirichlet form, and obtain a necessary and 
sufficient condition for it to be transient.  We show that 
the tail $\sigma$--field of the diffusion is always trivial
and draw the usual conclusion that bounded space--time 
harmonic functions are constant. In the transient case,
we identify the Martin compactification and  obtain the 
corresponding integral representations of excessive and
harmonic functions.  Using Ray--Knight methods, 
we show that the only entrance laws for the diffusion are 
the trivial ones that arise from starting the process inside
the state--space.  Finally, we use the Dirichlet form
stochastic calculus to obtain a semimartingale description 
of the diffusion that involves local time additive
functionals associated with each branch point of the tree.

evans@stat.berkeley.edu

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1299. LARGE DEVIATIONS OF JACKSON NETWORKS

Irina Ignatiouk-Robert

The problem of  large deviations for a Jackson  network is analyzed in
detail.   A new representation  of the  rate function  is given  and a
simple procedure is  proposed to get its closed  form expression.  The
methods used rely on twisted distributions, localized processes, fluid
limits and a careful analysis of some functions.

Irina.Ignatiouk@u-cergy.fr

1300. MEAN-FIELD LATTICE TREES

Christian Borgs, Jennifer Chayes, Remco van der Hofstad and Gordon Slade

We introduce a mean-field model of lattice trees based on 
embeddings into Z^d of abstract trees having a critical Poisson offspring
distribution.  This model provides a combinatorial
interpretation for the self-consistent mean-field model introduced 
previously by Derbez and Slade, and provides an alternate approach
to work of Aldous.   The scaling limit of the mean-field
model is integrated super-Brownian excursion (ISE), in all dimensions.
We also introduce a model of weakly self-avoiding lattice trees,
in which an embedded tree receives a penalty e^{-\beta} for each
self-intersection.  The weakly self-avoiding lattice trees provide
a natural interpolation between the mean-field model (\beta =0),
and the usual model of strictly self-avoiding lattice trees (\beta =\infty)
which associates the uniform measure to the set of lattice trees of
the same size.

slade@math.mcmaster.ca

1301. LATTICE TREES, PERCOLATION AND SUPER-BROWNIAN MOTION

Gordon Slade

This paper surveys the results of recent collaborations with
Eric Derbez and with Takashi Hara, which show that integrated
super-Brownian excursion (ISE) arises as the scaling limit of both
lattice trees and the incipient infinite percolation cluster, 
in high dimensions.  A potential extension to oriented percolation
is also mentioned.

slade@math.mcmaster.ca

1302. THE SCALING LIMIT OF THE INCIPIENT INFINITE CLUSTER IN HIGH-DIMENSIONAL PERCOLATION. I. CRITICAL EXPONENTS

Takashi Hara and Gordon Slade

This is the first of two papers on the critical behaviour
of bond percolation models in high dimensions.  In this paper,
we obtain strong joint control of the critical exponents
\eta and \delta, for the nearest-neighbour model in very
high dimensions d \gg 6 and for sufficiently spread-out models
in all dimensions d>6.  The exponent \eta describes the low frequency 
behaviour of the Fourier transform of the critical two-point connectivity 
function, while \delta describes the behaviour of the magnetization at 
the critical point.  Our main result is an asymptotic relation
showing that, in a joint sense, \eta = 0 and \delta = 2.
The proof uses a major extension of our earlier expansion 
method for percolation.  This result provides evidence that the 
scaling limit of the incipient infinite cluster is the random 
probability measure on R^d known as integrated
super-Brownian excursion (ISE), in dimensions above 6.
In the sequel to this paper, we extend our methods to
prove that the scaling limits of the incipient infinite cluster's 
two-point and three-point functions are those 
of ISE for the nearest-neighbour model in dimensions d \gg 6.

slade@math.mcmaster.ca

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1303. THE SCALING LIMIT OF THE INCIPIENT INFINITE CLUSTER IN HIGH-DIMENSIONAL PERCOLATION. II. INTEGRATED SUPER-BROWNIAN EXCURSION

Takashi Hara and Gordon Slade

For independent nearest-neighbour bond percolation on Z^D with d \gg 6,
we prove that the incipient infinite cluster's two-point function and 
three-point function converge to those of integrated super-Brownian 
excursion (ISE) in the scaling limit.   The proof is based on an extension 
of the new expansion for percolation derived in a previous paper,
and involves treating the magnetic field as a complex variable.
A special case of our result for the two-point function implies that the
probability that the cluster of the origin consists of n sites, at the 
critical point, is given by a multiple of n^{-3/2}, plus an error term 
of order n^{-3/2-\epsilon} with \epsilon >0.  This is a strong statement
that the critical exponent \delta is given by \delta =2.

slade@math.mcmaster.ca

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1304. CONVERGENCE OF SCALED RENEWAL PROCESSES TO FRACTIONAL BROWNIAN MOTION

Ingemar Kaj

The superposition process of independent counting renewal 
processes associated with a heavy-tailed interarrival time 
distribution is shown to converge weakly after rescaling in 
time and space to fractional Brownian motion, as the number 
of renewal processes tends to infinity. Corresponding 
results for continuous arrival fluid processes are 
discussed.

ikaj@math.uu.se

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1305. ECCENTRIC BEHAVIORS OF THE BROWNIAN SHEET ALONG LINES

Robert C. Dalang and T. Mountford

Distinct excursion intervals of a Brownian motion (that 
correspond to a fixed level) have no common endpoints. 
What is the situation for distinct excursion sets of a
Brownian sheet? These sets are termed Brownian bubbles
in the literature, and this paper examines how bubbles 
from fixed or random levels come into contact with each 
other, by examining whether or not the Brownian sheet 
restricted to a specific type of curve can have a point 
of increase. At random levels, we show that points of 
increase can occur along horizontal lines, while at 
fixed levels, such a point of increase can occur at the 
corner of a broken line segment with a right-angle.
In addition, the Hausdorff dimension of the set of 
points with this last property is shown to be 1/2 a.s.

robert.dalang@epfl.ch  malloy@math.ucla.edu

1306. MULTIPARAMETER RANDOM FIELDS

George G. Zhyrnyy

The dissertation is devoted to investigation of
martingale properties of multiparameter random fields with
applications to statistical problems. New approach to study
properties of strong martingales without Cairoli-Walsh
condition is developed. This approach allows us to use many
properties of strong martingales that has been already
proved under Cairoli-Walsh condition. Effectiveness of
method was verified while studying statistical properties
of random fields, averaging in stochastic equations, etc.

george@skif.net

1307. AN APPROACH TO THE EXISTENCE OF UNIQUE INVARIANT PROBABILITIES FOR MARKOV PROCESSES

Rabi Bhattacharya and Ed Waymire

A notion of localized splitting is introduced as a
further extension of the splitting notions for iterated
monotone maps introduced earlier by Dubins and Freedman (1966)
and more generally by Bhattacharya and Majumdar (1999). We
will see that under quite general condtions, localized
splitting theory is a natural extension of the Doeblin (1937)
minorization theory, Harris (1956) recurrence theory, 
splitting theory of Nummelin (1978) and regeneration theory
of Athreya and Ney (1978), under which we can prove the
existence a a unique invariant probability. By way of
introduction we also provide a natural coupling proof of 
the ergodic problem for Markov processes on general state
space under Doeblin's minorization condition which heretofore
seems to have gone unnoticed. The paper is concluded with
some new applications of splitting theory to random iterated
quadratic maps.

bhattach@ucs.indiana.edu waymire@math.orst.edu

1308. A SIMPLE APPROACH TO GLOBAL REGIME OF THE RANDOM MATRIX THEORY

Leonid Pastur

We discuss a method of the asymptotic computation of moments of the
normalized eigenvalue counting measure of random matrices of large order. The
method is based on the resolvent identity and on some formulas relating
expectations of certain matrix functions and the expectations including their
derivatives or, equivalently, on some simple formulas of the perturbation
theory. In the framework of this unique approach we obtain functional equations
for the Stieltjes transforms of the limiting normalized eigenvalue counting
measure and the bounds for the rate of convergence for the majority known
random matrix ensembles.

pastur@math.jussieu.fr

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1309. SPARSE RANDOM MATRICES AND STATISTICS OF ROOTED TREES

A. Khorunzhy

We consider the ensemble of N-dimensional random symmetric matrices A that
have, in average, p non-zero elements per row. We study the asymptotic behavior
of the norm of A in the limit of infinitely increasing N and p. We prove that
the value p= log N is the critical one for the norm to be either bounded or
not. The arguments are based on the calculus of the tree-type graphs.
Asymptotic properties of sparse random matrices essentially depend on the
typical degree of a tree vertex that we show to be finite.

khorunjy@math.jussieu.fr

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1310. SAMPLE PATH LARGE DEVIATIONS FOR A CLASS OF MARKOV CHAINS RELATED TO DISORDERED MEAN FIELD MODELS

Anton Bovier, Veronique Gayrard

We prove a large deviation principle on path space for a class of discrete
time Markov processes whose state space is the intersection of a regular domain
$\L\subset \R^d$ with some lattice of spacing $\e$. Transitions from $x$ to $y$
are allowed if $\e^{-1}(x-y)\in \D$ for some fixed set of vectors $\D$. The
transition probabilities $p_\e(t,x,y)$, which themselves depend on $\e$, are
allowed to depend on the starting point $x$ and the time $t$ in a sufficiently
regular way, except near the boundaries, where some singular behaviour is
allowed. The rate function is identified as an action functional which is given
as the integral of a Lagrange function. %of time dependent relativistic
classical mechanics. Markov processes of this type arise in the study of mean
field dynamics of disordered mean field models.

bovier@hilbert.wias-berlin.de

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1311. ALGEBRAIC ASPECTS OF INCREASING SUBSEQUENCES

Jinho Baik and Eric M. Rains

We present a number of results relating partial Cauchy-Littlewood sums,
integrals over the compact classical groups, and increasing subsequences of
permutations. These include: integral formulae for the distribution of the
longest increasing subsequence of a random involution with constrained number
of fixed points; new formulae for partial Cauchy-Littlewood sums, as well as
new proofs of old formulae; relations of these expressions to orthogonal
polynomials on the unit circle; and explicit bases for invariant spaces of the
classical groups, together with appropriate generalizations of the
straightening algorithm.

rains@research.att.com

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1312. THE ASYMPTOTICS OF MONOTONE SUBSEQUENCES OF INVOLUTIONS

Jinho Baik and Eric M. Rains

We consider the asymptotics of the distributions of the lengths of the
longest monotone subsequences of random involutions taken from various
ensembles. They are ensembles of involutions, signed involutions, involutions
with constraint on the number of fixed points and signed involutions with
constraint on the number of fixed and negated points. The resulting limiting
distributions are expressed in terms of either the Tracy-Widom distributions
for the largest eigenvalues of random GOE, GUE, GSE matrices, or new classes of
distributions defined in terms of the solution of the Riemann-Hilbert problem
for the Painlev\'e II equation. These new classes of distributions interpolate
between certain pairs of the Tracy-Widom distributions. We also consider the
asymptotic distributions of the length of the second rows of the corresponding
Young diagram ensembles. In each case, convergence of moments is also obtained.

baik@cims.nyu.edu

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1313. COMBINATORICS OF FREE CUMULANTS

Bernadette Krawczyk, Roland Speicher

We derive a formula for expressing free cumulants whose entries are products
of random variables in terms of the lattice structure of non-crossing
partitions. We show the usefulness of that result by giving direct and
conceptually simple proofs for a lot of results about $R$-diagonal elements.
Our investigations do not assume the trace property for the considered linear
functionals.

f58@ix.urz.uni-heidelberg.de

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1314. ON UNIVERSALITY OF SMOOTHED EIGENVALUE DENSITY OF LARGE RANDOM MATRICES

A. Boutet de Monvel, A. Khorunzhy

We describe the resolvent approach for the rigorous study of the mescoscopic
regime of Hermitian matrix spectra. We present results reflecting the universal
behavior of the smoothed density of eigenvalue distribution of large random
matrices

khorunjy@math.jussieu.fr (Aliosha Khorunjy)

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1315. THE LIL FOR CANONICAL U-STATISTICS OF ORDER 2

Evarist Gin\'e, Stanis{\l}aw Kwapie\'n, Rafa{\l} Lata{\l}a 
 and Joel Zinn

Let X,X_1,X_2,... be independent identically distributed random variables and
let h(x,y)=h(y,x) be a measurable function of two variables. It is shown that
the bounded law of the iterated logarithm, $\limsup_n (n\log\log
n)^{-1}|\sum_{1<= i< j<= n}h(X_i,X_j)|<\infty$ a.s., holds if and only if the
following three conditions are satisfied: h is canonical for the law of X (that
is Eh(X,y)=0 for almost y) and there exists $C<\infty$ such that, both,
$E\min(h^2(X_1,X_2),u)<C\log\log u$ for all large u and
$sup\{Eh(X_1,X_2)f(X_1)g(X_2):|f(X)|_2<1,\|g(X)\|_2<1, \|f\|_\infty<\infty,
\|g\|_\infty<\infty\}< C$.

rlatala@mimuw.edu.pl

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1316. STATIONARY MEASURES FOR RANDOM WALKS IN A RANDOM ENVIRONMENT WITH RANDOM SCENERY

Russell Lyons and Oded Schramm

Let $\Gamma$ act on a countable set V with only finitely many orbits. Given a
$\Gamma$-invariant random environment for a Markov chain on V and a random
scenery, we exhibit, under certain conditions, an equivalent stationary measure
for the environment and scenery from the viewpoint of the random walker. Such
theorems have been very useful in investigations of percolation on
quasi-transitive graphs.

rdlyons@indiana.edu

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1317. Z-MEASURES ON PARTITIONS, ROBINSON-SCHENSTED-KNUTH CORRESPONDENCE, AND BETA=2 RANDOM MATRIX ENSEMBLES

Alexei Borodin and Grigori Olshanski

We suggest an hierarchy of all the results known so far about the connection
of the asymptotics of combinatorial or representation theoretic problems with
``beta=2 ensembles'' arising in the random matrix theory. We show that all such
results are, essentially, degenerations of one general situation arising from
so-called generalized regular representations of the infinite symmetric group.

borodine@math.upenn.edu

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1318. STATISTICAL PROPERTIES OF BRAID GROUPS IN LOCALLY FREE APPROXIMATION

A.M. Vershik, S. Nechaev, R. Bikbov

Statistical and probabilistic characteristics of locally free group with
growing number of generators are defined and their application to statistics of
braid groups is given.

nechaev@ipno.in2p3.fr

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1319. ON THE EFFECT OF ADDING EPSILON-BERNOULLI PERCOLATION TO EVERYWHERE PERCOLATING SUBGRAPHS OF Z^D

Itai Benjamini, Olle Haggstrom and Oded Schramm

We show that adding epsilon-Bernoulli percolation to an everywhere
percolating subgraph of Z^2 results in a graph which has large scale geometry
similar to that of supercritical Bernoulli percolation, in various specific
senses. We conjecture similar behavior in higher dimensions.

itai@wisdom.weizmann.ac.il

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1320. SEPARATION OF VARIABLES IN THE KRAMERS EQUATION

Renat Zhdanov and Alexander Zhalij

We consider the problem of separation of variables in the Kramers equation
admitting a non-trivial symmetry group. Provided the external potential $V(x)$
is at most quadratic, a complete solution of the problem of separation of
variables is obtained. Furthermore, we construct solutions of the Kramers
equation with separated variables in explicit form.

zhaliy@imath.kiev.ua

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1321. TAYLOR-KUBO FORMULA FOR TURBULENT DIFFUSION IN A NON-MIXING FLOW WITH LONG-RANGE CORRELATION

Albert C. Fannjiang and Tomasz Komorowski

We prove the Taylor-Kubo formula for a class of isotropic, non-mixing flows
with long-range correlation.
For the proof, we develop the method of high order correctors expansion.

fannjian@math.ucdavis.edu

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1322. FRACTIONAL BROWNIAN MOTION LIMIT FOR A MODEL OF TURBULENT TRANSPORT

Albert Fannjiang, Tomasz Komorowski 

Passive scalar motion in a family of random Gaussian velocity fields with
long-range correlations is shown to converge to persistent fractional Brownian
motions in long times.

fannjian@math.ucdavis.edu

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1323. DISCRETE ORTHOGONAL POLYNOMIAL ENSEMBLES AND THE PLANCHEREL MEASURE

Kurt Johansson

We consider discrete orthogonal polynomial ensembles which are discrete
analogues of the orthogonal polynomial ensembles in random matrix theory. These
ensembles occur in certain problems in combinatorial probability and can be
thought of as probability measures on partitions. The Meixner ensemble is
related to a two-dimensional directed growth model, and the Charlier ensemble
is related to the lengths of weakly increasing subsequences in random words.
The Krawtchouk ensemble occurs in connection with zig-zag paths in randon
domino tilings of the Aztec diamond, and also in a certain simplified directed
first-passage percolation model. We use the Charlier ensemble to investigate
the asymptotics of the length of the longest weakly increasing subsequence in a
random word and to prove a conjecture of Tracy and Widom. As a limit of the
Meixner ensemble or the Charlier ensemble we obtain the Plancherel measure on
partitions, and using this we prove a conjecture of Baik, Deift and Johansson
that under the Plancherel measure, the joint distribution of the lengths of the
first k rows in the partition, appropriately scaled, converges to the
asymptotic joint distribution for the k largest eigenvalues of a random matrix
from the Gaussian Unitary Ensemble. In this problem a certain discrete kernel,
which we call the discrete Bessel kernel, plays an important role.

kurtj@math.kth.se

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